Certification Problem
Input (COPS 6)
We consider the TRS containing the following rules:
|
f(f(x,y),z) |
→ |
f(x,f(y,z)) |
(1) |
|
f(i(x1),x1) |
→ |
e |
(2) |
The underlying signature is as follows:
{f/2, i/1, e/0}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2023)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(f(i(f5),f5),f6) |
|
→
|
f(e,f6) |
|
= |
t1
|
| t0
|
= |
f(f(i(f5),f5),f6) |
|
→
|
f(i(f5),f(f5,f6)) |
|
= |
t1
|
The two resulting terms cannot be joined for the following reason:
-
The reachable terms of these two terms are approximated via the following two tree automata,
and the tree automata have an empty intersection.
-
Automaton 1
-
final states:
{1}
-
transitions:
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
-
Automaton 2
-
final states:
{4}
-
transitions:
| f5 |
→ |
59 |
| f6 |
→ |
5 |
|
i(59) |
→ |
9 |
|
f(59,5) |
→ |
7 |
|
f(9,7) |
→ |
4 |
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
| 59 |
» |
59 |
| 9 |
» |
9 |
| 5 |
» |
5 |
| 7 |
» |
7 |
| 4 |
» |
4 |